(* *)
(**************************************************************************)
-include "ground_2/steps/rtc_ist_shift.ma".
-include "ground_2/steps/rtc_ist_plus.ma".
-include "ground_2/steps/rtc_ist_max.ma".
+include "ground/xoa/ex_4_3.ma".
+include "ground/steps/rtc_ist_shift.ma".
+include "ground/steps/rtc_ist_plus.ma".
+include "ground/steps/rtc_ist_max.ma".
+include "static_2/syntax/sh.ma".
include "basic_2/notation/relations/pty_6.ma".
include "basic_2/rt_transition/cpg.ma".
(* T-BOUND CONTEXT-SENSITIVE PARALLEL T-TRANSITION FOR TERMS ****************)
definition cpt (h) (G) (L) (n): relation2 term term ≝
- λT1,T2. â\88\83â\88\83c. ð\9d\90\93â¦\83n,câ¦\84 & â¦\83G,Lâ¦\84 â\8a¢ T1 â¬\88[eq â\80¦,c,h] T2.
+ λT1,T2. â\88\83â\88\83c. ð\9d\90\93â\9dªn,câ\9d« & â\9dªG,Lâ\9d« â\8a¢ T1 â¬\88[sh_is_next h,eq â\80¦,c] T2.
interpretation
"t-bound context-sensitive parallel t-transition (term)"
(* Basic properties *********************************************************)
lemma cpt_ess (h) (G) (L):
- â\88\80s. â¦\83G,Lâ¦\84 ⊢ ⋆s ⬆[h,1] ⋆(⫯[h]s).
-/2 width=3 by cpg_ess, ex2_intro/ qed.
+ â\88\80s. â\9dªG,Lâ\9d« ⊢ ⋆s ⬆[h,1] ⋆(⫯[h]s).
+/3 width=3 by cpg_ess, ex2_intro/ qed.
lemma cpt_delta (h) (n) (G) (K):
- â\88\80V1,V2. â¦\83G,Kâ¦\84 ⊢ V1 ⬆[h,n] V2 →
- ∀W2. ⇧*[1] V2 ≘ W2 → ⦃G,K.ⓓV1⦄ ⊢ #0 ⬆[h,n] W2.
+ â\88\80V1,V2. â\9dªG,Kâ\9d« ⊢ V1 ⬆[h,n] V2 →
+ ∀W2. ⇧[1] V2 ≘ W2 → ❪G,K.ⓓV1❫ ⊢ #0 ⬆[h,n] W2.
#h #n #G #K #V1 #V2 *
/3 width=5 by cpg_delta, ex2_intro/
qed.
lemma cpt_ell (h) (n) (G) (K):
- â\88\80V1,V2. â¦\83G,Kâ¦\84 ⊢ V1 ⬆[h,n] V2 →
- ∀W2. ⇧*[1] V2 ≘ W2 → ⦃G,K.ⓛV1⦄ ⊢ #0 ⬆[h,↑n] W2.
+ â\88\80V1,V2. â\9dªG,Kâ\9d« ⊢ V1 ⬆[h,n] V2 →
+ ∀W2. ⇧[1] V2 ≘ W2 → ❪G,K.ⓛV1❫ ⊢ #0 ⬆[h,↑n] W2.
#h #n #G #K #V1 #V2 *
/3 width=5 by cpg_ell, ex2_intro, ist_succ/
qed.
lemma cpt_lref (h) (n) (G) (K):
- â\88\80T,i. â¦\83G,Kâ¦\84 â\8a¢ #i â¬\86[h,n] T â\86\92 â\88\80U. â\87§*[1] T ≘ U →
- â\88\80I. â¦\83G,K.â\93\98{I}â¦\84 ⊢ #↑i ⬆[h,n] U.
+ â\88\80T,i. â\9dªG,Kâ\9d« â\8a¢ #i â¬\86[h,n] T â\86\92 â\88\80U. â\87§[1] T ≘ U →
+ â\88\80I. â\9dªG,K.â\93\98[I]â\9d« ⊢ #↑i ⬆[h,n] U.
#h #n #G #K #T #i *
/3 width=5 by cpg_lref, ex2_intro/
qed.
lemma cpt_bind (h) (n) (G) (L):
- â\88\80V1,V2. â¦\83G,Lâ¦\84 â\8a¢ V1 â¬\86[h,0] V2 â\86\92 â\88\80I,T1,T2. â¦\83G,L.â\93\91{I}V1â¦\84 ⊢ T1 ⬆[h,n] T2 →
- â\88\80p. â¦\83G,Lâ¦\84 â\8a¢ â\93\91{p,I}V1.T1 â¬\86[h,n] â\93\91{p,I}V2.T2.
+ â\88\80V1,V2. â\9dªG,Lâ\9d« â\8a¢ V1 â¬\86[h,0] V2 â\86\92 â\88\80I,T1,T2. â\9dªG,L.â\93\91[I]V1â\9d« ⊢ T1 ⬆[h,n] T2 →
+ â\88\80p. â\9dªG,Lâ\9d« â\8a¢ â\93\91[p,I]V1.T1 â¬\86[h,n] â\93\91[p,I]V2.T2.
#h #n #G #L #V1 #V2 * #cV #HcV #HV12 #I #T1 #T2 *
/3 width=5 by cpg_bind, ist_max_O1, ex2_intro/
qed.
lemma cpt_appl (h) (n) (G) (L):
- â\88\80V1,V2. â¦\83G,Lâ¦\84 ⊢ V1 ⬆[h,0] V2 →
- â\88\80T1,T2. â¦\83G,Lâ¦\84 â\8a¢ T1 â¬\86[h,n] T2 â\86\92 â¦\83G,Lâ¦\84 ⊢ ⓐV1.T1 ⬆[h,n] ⓐV2.T2.
+ â\88\80V1,V2. â\9dªG,Lâ\9d« ⊢ V1 ⬆[h,0] V2 →
+ â\88\80T1,T2. â\9dªG,Lâ\9d« â\8a¢ T1 â¬\86[h,n] T2 â\86\92 â\9dªG,Lâ\9d« ⊢ ⓐV1.T1 ⬆[h,n] ⓐV2.T2.
#h #n #G #L #V1 #V2 * #cV #HcV #HV12 #T1 #T2 *
/3 width=5 by ist_max_O1, cpg_appl, ex2_intro/
qed.
lemma cpt_cast (h) (n) (G) (L):
- â\88\80U1,U2. â¦\83G,Lâ¦\84 ⊢ U1 ⬆[h,n] U2 →
- â\88\80T1,T2. â¦\83G,Lâ¦\84 â\8a¢ T1 â¬\86[h,n] T2 â\86\92 â¦\83G,Lâ¦\84 ⊢ ⓝU1.T1 ⬆[h,n] ⓝU2.T2.
+ â\88\80U1,U2. â\9dªG,Lâ\9d« ⊢ U1 ⬆[h,n] U2 →
+ â\88\80T1,T2. â\9dªG,Lâ\9d« â\8a¢ T1 â¬\86[h,n] T2 â\86\92 â\9dªG,Lâ\9d« ⊢ ⓝU1.T1 ⬆[h,n] ⓝU2.T2.
#h #n #G #L #U1 #U2 * #cU #HcU #HU12 #T1 #T2 *
/3 width=6 by cpg_cast, ex2_intro/
qed.
lemma cpt_ee (h) (n) (G) (L):
- â\88\80U1,U2. â¦\83G,Lâ¦\84 â\8a¢ U1 â¬\86[h,n] U2 â\86\92 â\88\80T. â¦\83G,Lâ¦\84 ⊢ ⓝU1.T ⬆[h,↑n] U2.
+ â\88\80U1,U2. â\9dªG,Lâ\9d« â\8a¢ U1 â¬\86[h,n] U2 â\86\92 â\88\80T. â\9dªG,Lâ\9d« ⊢ ⓝU1.T ⬆[h,↑n] U2.
#h #n #G #L #V1 #V2 *
/3 width=3 by cpg_ee, ist_succ, ex2_intro/
qed.
-(* Basic properties *********************************************************)
-
lemma cpt_refl (h) (G) (L): reflexive … (cpt h G L 0).
/3 width=3 by cpg_refl, ex2_intro/ qed.
+(* Advanced properties ******************************************************)
+
+lemma cpt_sort (h) (G) (L):
+ ∀n. n ≤ 1 → ∀s. ❪G,L❫ ⊢ ⋆s ⬆[h,n] ⋆((next h)^n s).
+#h #G #L * //
+#n #H #s <(le_n_O_to_eq n) /2 width=1 by le_S_S_to_le/
+qed.
+
(* Basic inversion lemmas ***************************************************)
lemma cpt_inv_atom_sn (h) (n) (J) (G) (L):
- â\88\80X2. â¦\83G,Lâ¦\84 â\8a¢ â\93ª{J} ⬆[h,n] X2 →
- ∨∨ ∧∧ X2 = ⓪{J} & n = 0
+ â\88\80X2. â\9dªG,Lâ\9d« â\8a¢ â\93ª[J] ⬆[h,n] X2 →
+ ∨∨ ∧∧ X2 = ⓪[J] & n = 0
| ∃∃s. X2 = ⋆(⫯[h]s) & J = Sort s & n =1
- | â\88\83â\88\83K,V1,V2. â¦\83G,Kâ¦\84 â\8a¢ V1 â¬\86[h,n] V2 & â\87§*[1] V2 ≘ X2 & L = K.ⓓV1 & J = LRef 0
- | â\88\83â\88\83m,K,V1,V2. â¦\83G,Kâ¦\84 â\8a¢ V1 â¬\86[h,m] V2 & â\87§*[1] V2 ≘ X2 & L = K.ⓛV1 & J = LRef 0 & n = ↑m
- | â\88\83â\88\83I,K,T,i. â¦\83G,Kâ¦\84 â\8a¢ #i â¬\86[h,n] T & â\87§*[1] T â\89\98 X2 & L = K.â\93\98{I} & J = LRef (↑i).
+ | â\88\83â\88\83K,V1,V2. â\9dªG,Kâ\9d« â\8a¢ V1 â¬\86[h,n] V2 & â\87§[1] V2 ≘ X2 & L = K.ⓓV1 & J = LRef 0
+ | â\88\83â\88\83m,K,V1,V2. â\9dªG,Kâ\9d« â\8a¢ V1 â¬\86[h,m] V2 & â\87§[1] V2 ≘ X2 & L = K.ⓛV1 & J = LRef 0 & n = ↑m
+ | â\88\83â\88\83I,K,T,i. â\9dªG,Kâ\9d« â\8a¢ #i â¬\86[h,n] T & â\87§[1] T â\89\98 X2 & L = K.â\93\98[I] & J = LRef (↑i).
#h #n #J #G #L #X2 * #c #Hc #H
elim (cpg_inv_atom1 … H) -H *
[ #H1 #H2 destruct /3 width=1 by or5_intro0, conj/
-| #s #H1 #H2 #H3 destruct /3 width=3 by or5_intro1, ex3_intro/
+| #s1 #s2 #H1 #H2 #H3 #H4 destruct /3 width=3 by or5_intro1, ex3_intro/
| #cV #K #V1 #V2 #HV12 #HVT2 #H1 #H2 #H3 destruct
/4 width=6 by or5_intro2, ex4_3_intro, ex2_intro/
| #cV #K #V1 #V2 #HV12 #HVT2 #H1 #H2 #H3 destruct
]
qed-.
+lemma cpt_inv_sort_sn (h) (n) (G) (L) (s):
+ ∀X2. ❪G,L❫ ⊢ ⋆s ⬆[h,n] X2 →
+ ∧∧ X2 = ⋆(((next h)^n) s) & n ≤ 1.
+#h #n #G #L #s #X2 * #c #Hc #H
+elim (cpg_inv_sort1 … H) -H * #H1 #H2 destruct
+[ /2 width=1 by conj/
+| #H1 #H2 destruct /2 width=1 by conj/
+]
+qed-.
+
+lemma cpt_inv_zero_sn (h) (n) (G) (L):
+ ∀X2. ❪G,L❫ ⊢ #0 ⬆[h,n] X2 →
+ ∨∨ ∧∧ X2 = #0 & n = 0
+ | ∃∃K,V1,V2. ❪G,K❫ ⊢ V1 ⬆[h,n] V2 & ⇧[1] V2 ≘ X2 & L = K.ⓓV1
+ | ∃∃m,K,V1,V2. ❪G,K❫ ⊢ V1 ⬆[h,m] V2 & ⇧[1] V2 ≘ X2 & L = K.ⓛV1 & n = ↑m.
+#h #n #G #L #X2 * #c #Hc #H elim (cpg_inv_zero1 … H) -H *
+[ #H1 #H2 destruct /4 width=1 by ist_inv_00, or3_intro0, conj/
+| #cV #K #V1 #V2 #HV12 #HVT2 #H1 #H2 destruct
+ /4 width=8 by or3_intro1, ex3_3_intro, ex2_intro/
+| #cV #K #V1 #V2 #HV12 #HVT2 #H1 #H2 destruct
+ elim (ist_inv_plus_SO_dx … H2) -H2 // #m #Hc #H destruct
+ /4 width=8 by or3_intro2, ex4_4_intro, ex2_intro/
+]
+qed-.
+
+lemma cpt_inv_zero_sn_unit (h) (n) (I) (K) (G):
+ ∀X2. ❪G,K.ⓤ[I]❫ ⊢ #0 ⬆[h,n] X2 → ∧∧ X2 = #0 & n = 0.
+#h #n #I #G #K #X2 #H
+elim (cpt_inv_zero_sn … H) -H *
+[ #H1 #H2 destruct /2 width=1 by conj/
+| #Y #X1 #X2 #_ #_ #H destruct
+| #m #Y #X1 #X2 #_ #_ #H destruct
+]
+qed.
+
+lemma cpt_inv_lref_sn (h) (n) (G) (L) (i):
+ ∀X2. ❪G,L❫ ⊢ #↑i ⬆[h,n] X2 →
+ ∨∨ ∧∧ X2 = #(↑i) & n = 0
+ | ∃∃I,K,T. ❪G,K❫ ⊢ #i ⬆[h,n] T & ⇧[1] T ≘ X2 & L = K.ⓘ[I].
+#h #n #G #L #i #X2 * #c #Hc #H elim (cpg_inv_lref1 … H) -H *
+[ #H1 #H2 destruct /4 width=1 by ist_inv_00, or_introl, conj/
+| #I #K #V2 #HV2 #HVT2 #H destruct
+ /4 width=6 by ex3_3_intro, ex2_intro, or_intror/
+]
+qed-.
+
+lemma cpt_inv_lref_sn_ctop (h) (n) (G) (i):
+ ∀X2. ❪G,⋆❫ ⊢ #i ⬆[h,n] X2 → ∧∧ X2 = #i & n = 0.
+#h #n #G * [| #i ] #X2 #H
+[ elim (cpt_inv_zero_sn … H) -H *
+ [ #H1 #H2 destruct /2 width=1 by conj/
+ | #Y #X1 #X2 #_ #_ #H destruct
+ | #m #Y #X1 #X2 #_ #_ #H destruct
+ ]
+| elim (cpt_inv_lref_sn … H) -H *
+ [ #H1 #H2 destruct /2 width=1 by conj/
+ | #Z #Y #X0 #_ #_ #H destruct
+ ]
+]
+qed.
+
+lemma cpt_inv_gref_sn (h) (n) (G) (L) (l):
+ ∀X2. ❪G,L❫ ⊢ §l ⬆[h,n] X2 → ∧∧ X2 = §l & n = 0.
+#h #n #G #L #l #X2 * #c #Hc #H elim (cpg_inv_gref1 … H) -H
+#H1 #H2 destruct /2 width=1 by conj/
+qed-.
+
lemma cpt_inv_bind_sn (h) (n) (p) (I) (G) (L) (V1) (T1):
- â\88\80X2. â¦\83G,Lâ¦\84 â\8a¢ â\93\91{p,I}V1.T1 ⬆[h,n] X2 →
- â\88\83â\88\83V2,T2. â¦\83G,Lâ¦\84 â\8a¢ V1 â¬\86[h,0] V2 & â¦\83G,L.â\93\91{I}V1â¦\84 ⊢ T1 ⬆[h,n] T2
- & X2 = ⓑ{p,I}V2.T2.
+ â\88\80X2. â\9dªG,Lâ\9d« â\8a¢ â\93\91[p,I]V1.T1 ⬆[h,n] X2 →
+ â\88\83â\88\83V2,T2. â\9dªG,Lâ\9d« â\8a¢ V1 â¬\86[h,0] V2 & â\9dªG,L.â\93\91[I]V1â\9d« ⊢ T1 ⬆[h,n] T2
+ & X2 = ⓑ[p,I]V2.T2.
#h #n #p #I #G #L #V1 #T1 #X2 * #c #Hc #H
elim (cpg_inv_bind1 … H) -H *
[ #cV #cT #V2 #T2 #HV12 #HT12 #H1 #H2 destruct
qed-.
lemma cpt_inv_appl_sn (h) (n) (G) (L) (V1) (T1):
- â\88\80X2. â¦\83G,Lâ¦\84 ⊢ ⓐV1.T1 ⬆[h,n] X2 →
- â\88\83â\88\83V2,T2. â¦\83G,Lâ¦\84 â\8a¢ V1 â¬\86[h,0] V2 & â¦\83G,Lâ¦\84 ⊢ T1 ⬆[h,n] T2 & X2 = ⓐV2.T2.
+ â\88\80X2. â\9dªG,Lâ\9d« ⊢ ⓐV1.T1 ⬆[h,n] X2 →
+ â\88\83â\88\83V2,T2. â\9dªG,Lâ\9d« â\8a¢ V1 â¬\86[h,0] V2 & â\9dªG,Lâ\9d« ⊢ T1 ⬆[h,n] T2 & X2 = ⓐV2.T2.
#h #n #G #L #V1 #T1 #X2 * #c #Hc #H elim (cpg_inv_appl1 … H) -H *
[ #cV #cT #V2 #T2 #HV12 #HT12 #H1 #H2 destruct
elim (ist_inv_max … H2) -H2 #nV #nT #HcV #HcT #H destruct
qed-.
lemma cpt_inv_cast_sn (h) (n) (G) (L) (V1) (T1):
- â\88\80X2. â¦\83G,Lâ¦\84 ⊢ ⓝV1.T1 ⬆[h,n] X2 →
- â\88¨â\88¨ â\88\83â\88\83V2,T2. â¦\83G,Lâ¦\84 â\8a¢ V1 â¬\86[h,n] V2 & â¦\83G,Lâ¦\84 ⊢ T1 ⬆[h,n] T2 & X2 = ⓝV2.T2
- | â\88\83â\88\83m. â¦\83G,Lâ¦\84 ⊢ V1 ⬆[h,m] X2 & n = ↑m.
+ â\88\80X2. â\9dªG,Lâ\9d« ⊢ ⓝV1.T1 ⬆[h,n] X2 →
+ â\88¨â\88¨ â\88\83â\88\83V2,T2. â\9dªG,Lâ\9d« â\8a¢ V1 â¬\86[h,n] V2 & â\9dªG,Lâ\9d« ⊢ T1 ⬆[h,n] T2 & X2 = ⓝV2.T2
+ | â\88\83â\88\83m. â\9dªG,Lâ\9d« ⊢ V1 ⬆[h,m] X2 & n = ↑m.
#h #n #G #L #V1 #T1 #X2 * #c #Hc #H elim (cpg_inv_cast1 … H) -H *
[ #cV #cT #V2 #T2 #HV12 #HT12 #HcVT #H1 #H2 destruct
elim (ist_inv_max … H2) -H2 #nV #nT #HcV #HcT #H destruct
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